Question Number 187559 by Mingma last updated on 18/Feb/23
Answered by cortano12 last updated on 19/Feb/23
$$\:\frac{\mathrm{sin}\:{x}}{\mathrm{sin}\:{y}}\:.\frac{\mathrm{cos}\:{y}}{\mathrm{cos}\:{x}}\:=\:\frac{\mathrm{1}}{\mathrm{3}}\Rightarrow\frac{\mathrm{tan}\:{x}}{\mathrm{tan}\:{y}}=\frac{\mathrm{1}}{\mathrm{3}} \\ $$$$\:\mathrm{3tan}\:{x}=\mathrm{tan}\:{y} \\ $$$$\Rightarrow\mathrm{tan}\:\left({x}+{y}\right)=\frac{\mathrm{4tan}\:{x}}{\mathrm{1}−\mathrm{3tan}\:^{\mathrm{2}} {x}}\:=\pm\:\sqrt{\mathrm{15}} \\ $$$$\Rightarrow\mathrm{2sin}\:{x}=\mathrm{sin}\:{y}\:;\:\mathrm{2cos}\:{x}=\mathrm{3cos}\:{y} \\ $$$$\Rightarrow\mathrm{4}=\mathrm{sin}\:^{\mathrm{2}} {y}+\mathrm{9cos}\:^{\mathrm{2}} {y} \\ $$$$\Rightarrow\mathrm{4}=\mathrm{sin}\:^{\mathrm{2}} {y}+\mathrm{9}−\mathrm{9sin}\:^{\mathrm{2}} {y} \\ $$$$\Rightarrow\mathrm{sin}\:^{\mathrm{2}} {y}=\frac{\mathrm{5}}{\mathrm{8}}\:\wedge\:\mathrm{cos}\:^{\mathrm{2}} {y}=\frac{\mathrm{3}}{\mathrm{8}} \\ $$$$\Rightarrow\mathrm{tan}\:{y}=\pm\:\sqrt{\frac{\mathrm{5}}{\mathrm{3}}}\: \\ $$
Commented by Mingma last updated on 19/Feb/23
Great man!